Say we have a real-valued vector $x \in \mathbb{R}^p$. Now say we construct another vector $y$, where we take a small scalar $c>0$ away from each entry of $x$, where $c$ is smaller than each entry of $x$. We can bound the difference of the $\ell_2$ norms by
$$||y ||_2 -|| x||_2 \leq || y-x||_2 = ||-c||_2 =||c||_2$$.
However, as $||y ||_2 -|| x||_2 \leq 0$, the bound changes symbol. I wonder if we can find a bound from above such that it remains below 0?
The correct relation is $$ \Big|\|y\|_2-\|x\|_2\Big|\leq\|c\|_2 $$ so that $$ -\|c\|_2\leq\|y\|_2-\|x\|_2\leq\|c\|_2 $$